This has been asked and there are than one (paid) solutions to this problem.
In this article, I'm going to introduce a totally free method.
I ain't going to go into the steps but if you know how to do streaming to YouTube/FB, you will be able to understand this method easily. If you don't know how to do streaming to a single platform, you won't probably ask this question, right?
To make everything simple, the whole concept is to do streaming on a stream. I guess some people might already know how it goes, if you still don't get it, please read on.
First of all, you will use OBS to stream to either YouTube or Facebook as usual, let say you will use OBS to stream to YouTube first.
Now, you've a YouTube stream and now you will on to have it on Facebook live as well. To do that, you use a separate computer to play that YouTube stream. And then on this computer, start another OBS, capture that YouTube stream with OBS and then stream it to Facebook, QED.
That's what I meant by streaming a stream.